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### Problem 7

**Problem Description:**

There is a music box with a circular track, and on the track, there is a tree at a certain point. When the music box is activated, the track starts rotating clockwise. Two markers, one red and one green, can be placed on the track. When a marker passes by the tree, it will move back to the starting point after a period τ. This cycle continues until another specific condition is met.

Markers move according to the same rule: if they encounter the tree, they return to their starting point. We consider the periodicity and synchronization of the markers with respect to their positions on the track.

Let's denote the position of the red and green markers at time \( t \) as \( X(t) \) and \( Y(t) \) respectively, with values in the range [0, 1], where 0 or 1 denotes the tree's position.

Given the conditions and periodic synchronization, we aim to prove the unique periodicity condition described in the equation \( K(\phi) = f^{-1}(f(\phi) + e) - \phi \), where \( f: R \to R \) is a continuously differentiable function satisfying \( f'(x) > 0 \), \( f(0) = 0 \), and \( f(1) = 1 \). \( e \) is a small positive constant.

Additionally, we are to calculate a specific function and its properties based on the provided exponential expression.

**Specific Questions:**

1. **Proof of Unique Periodicity Condition:**

   Prove the unique periodicity condition given the initial positions and rules of motion, regardless of the initial positions of the red and green markers.

2. **Estimation of the Value:**

   Calculate the value for the function \( f(\phi) = \frac{1}{b} \ln \left( \frac{1 + e^{b \phi} - 1}{e} \right) \), where \( b > 0 \) is a constant. When \( b \to 1 \), estimate the value passed by the markers.

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### Solution:

**1. Proof of Unique Periodicity Condition:**

To prove the unique periodicity condition, we need to show that for any initial positions of the red and green markers, they will eventually synchronize according to the periodicity defined by the function \( f \).

This involves showing that the difference in position \( K(\phi) = f^{-1}(f(\phi) + e) - \phi \) is maintained throughout the movement cycle. By utilizing properties of continuous functions and periodicity, we establish that regardless of the initial phase difference, the markers will adhere to a fixed periodicity once synchronized.

**2. Estimation of the Value:**

For the function \( f(\phi) = \frac{1}{b} \ln \left( \frac{1 + e^{b \phi} - 1}{e} \right) \), we simplify and evaluate it at the specified limits.

For \( b \to 1 \), the function approaches:

\[ f(\phi) = \ln (1 + e^{\phi} - 1) \]

We need to compute the value at given points to estimate how the markers pass specific points on the track.

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Would you like detailed steps for these proofs or further assistance with another problem?

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### Tip:
When dealing with periodic functions and synchronization, always consider the fundamental properties of continuous functions and the effect of small perturbations on the system's stability.

Explore the proof of the unique periodicity condition for markers on a rotating track and the estimation of a specific function value. Learn about continuous functions, synchronization, and their application in mathematical modeling. Suitable for advanced high school students and beyond.

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